📚 AS AQA Philosophy: Formula & Theorem Quick Reference Handbook | AS AQA 哲学：公式定理速查手册
In philosophy, we do not find equations in the sense of E=mc², but we do encounter core principles, logical structures, and conceptual formulas that function like theorems. These underpin arguments, ethical systems, and metaphysical claims. This handbook collects the key ‘formulae’ you must know for AS AQA Philosophy, each presented as a crisp statement and then unpacked. Remember, a philosophical formula is a condensed insight that generates whole swathes of debate – own it, and you own the syllabus.
在哲学中，我们找不到 E=mc² 那样的方程，但确实会遇到核心原则、逻辑结构和概念性公式，它们就像定理一样运作。这些支撑着论证、伦理体系和形而上学主张。本手册汇集了 AS AQA 哲学必须掌握的关键“公式”，每一条先以精炼陈述呈现，再展开解释。记住，一个哲学公式就是浓缩的洞见，能派生出整片辩论疆域——掌握它们，你就掌握了整个考纲。
1. Descartes’ Cogito | 笛卡尔的我思故我在
Descartes’ foundational principle is often expressed as ‘Cogito, ergo sum’ – I think, therefore I am. Even if an evil demon deceives me about everything, the very act of being deceived requires my existence as a thinking thing. This formula establishes the indubitable bedrock of knowledge: one cannot coherently doubt one’s own existence while doubting.
笛卡尔的基础原则常被表述为“我思故我在”——我思考，所以我存在。即便有一个邪恶的恶魔在所有事情上欺骗我，被欺骗的这个行为本身就需要我作为一个思维之物而存在。这个公式确立了知识不可怀疑的基石：一个人在怀疑时，不可能前后一致地怀疑自己的存在。
Cogito, ergo sum. → Clear and distinct perception is the criterion of truth.
我思故我在。→ 清楚明晰的感知是真理的标准。
Descartes then builds his epistemology on the criterion that whatever is perceived clearly and distinctly must be true. This is the formula for rationalist certainty: from the cogito, we derive the existence of God and eventually the reliability of our senses. For AS, you must be ready to critique the Cartesian circle and the leap from ‘I think’ to ‘I am a thinking substance’.
随后，笛卡尔将其认识论建立在这样一个判定标准之上：凡是清楚、明晰地感知到的东西必定为真。这就是理性主义确定性的公式：从我思出发，推导出上帝的存在，最终推导出感官的可靠性。在 AS 阶段，你必须准备好批判笛卡尔循环，以及从“我思考”到“我是一个思维实体”的跳跃。
2. Hume’s Problem of Induction | 休谟的归纳问题
Hume’s formula can be stated as: ‘The future will resemble the past’ is neither a relation of ideas nor a matter of fact that can be rationally justified without circularity. All inductive reasoning assumes the uniformity of nature, yet we have no demonstrative or empirical proof for that uniformity that does not itself rely on induction.
休谟的公式可以表述为：“未来将类似于过去”既不是观念的关系，也不是可以不陷入循环而理性辩护的事实。所有归纳推理都预设了自然的齐一性，但我们对这一齐一性既无演绎证明也无经验证明，因为任何经验证明本身都依赖归纳。
Induction depends on the Principle of the Uniformity of Nature (PUN), which cannot be non-circularly justified. → Induction is a psychological habit, not a rationally grounded inference.
归纳依赖于自然齐一性原则（PUN），而该原则无法非循环地得到辩护。→ 归纳是一种心理习惯，而非理性奠基的推理。
Thus, the formula exposes a skeptical gap at the heart of empirical science. We cannot rationally infer cause and effect; we only observe constant conjunction. This ‘theorem’ is essential for any essay on scepticism about the external world or miracles.
因此，这个公式揭露了经验科学核心处的怀疑论裂隙。我们无法理性地推出因果关系；我们只能观察到恒常连结。这条“定理”对于任何关于外部世界或神迹的怀疑论论述都至关重要。
3. The Principle of Utility (Greatest Happiness Principle) | 效用原则（最大幸福原则）
Bentham’s formula: ‘It is the greatest happiness of the greatest number that is the measure of right and wrong.’ More formally, an action is right if it tends to produce the greatest net balance of pleasure over pain, considering all sentient beings affected.
边沁的公式：“最大多数人的最大幸福是衡量是非的标准。”更严格地说，一个行动如果在考虑所有受影响的众生之后，倾向于产生快乐减去痛苦的最大净余额，就是正确的。
Right action = maximise (∑ pleasure − ∑ pain) for all affected.
正确行动 = 最大化（∑ 快乐 − ∑ 痛苦）针对所有受影响者。
Mill refines this with qualitative distinctions, but the core formula remains a form of consequentialist calculus. For AS, you must know the difference between act and rule utilitarianism, and how the hedonic calculus (intensity, duration, certainty, etc.) attempts to operationalise this ethical theorem.
密尔用质的区分来细化它，但核心公式依然是后果主义计算的一种形式。在 AS 阶段，你必须知道行为功利主义和规则功利主义的区别，以及苦乐计算（强度、时长、确定性等）如何试图将这个伦理定理操作化。
4. Kant’s First Formulation of the Categorical Imperative | 康德定言命令的第一公式
Kant’s supreme moral formula: ‘Act only according to that maxim whereby you can at the same time will that it should become a universal law.’ This is the Formula of Universal Law (FUL), which serves as a test of moral permissibility.
康德的最高道德公式：“只按照你同时能够愿意它成为一个普遍法则的准则去行动。”这就是普遍法则公式（FUL），它充当道德可容许性的检验。
Maxim → Universalisation test: If a contradiction in conception or in will arises, the act is impermissible.
准则 → 普遍化检验：如果产生构想矛盾或意愿矛盾，则该行为不被允许。
Unlike utilitarian calculations, the categorical imperative is derived from pure practical reason. It is the central ‘equation’ of deontological ethics: the moral worth of an act lies in its being done for the sake of duty, not in the consequences. You must be able to apply the test to maxims such as ‘make a false promise’ and discuss its rigidity.
与功利主义计算不同，定言命令源于纯粹实践理性。它是道义论伦理学的中心“公式”：一个行动的道德价值在于它是出于义务而做出的，而不在于其后果。你必须能够将此检验应用于“做假承诺”之类的准则，并讨论其僵化性。
5. Anselm’s Ontological Argument (Definitional Formula) | 安瑟伦的本体论论证（定义公式）
Anselm’s formula works by definition: God is ‘that than which nothing greater can be conceived’. Even the fool who says ‘there is no God’ understands this definition. Existing in reality is greater than existing merely in the understanding. Therefore, if God existed only in the understanding, we could conceive of something greater (one that also exists in reality), which is a contradiction.
安瑟伦的公式通过定义起作用：上帝是“那无法设想有比之更伟大之物的东西”。即便是说“没有上帝”的愚人也理解这个定义。在现实中存在要比仅在理解中存在更伟大。因此，如果上帝只存在于理解中，我们就可以设想一个更伟大的东西（一个也在现实中存在的东西），这便是一个矛盾。
God = maximum conceivable greatness. → God must exist in reality, else not maximally great.
上帝 = 最大可设想之伟大。→ 上帝必然在现实中存在，否则就不是最大伟大。
Gaunilo’s island parody and Kant’s objection that ‘existence is not a predicate’ are built directly upon dissecting this formula. Your exam answer should unpack the hidden assumption that existence is a perfection or adds to greatness.
高尼罗的岛屿戏仿和康德的“存在不是谓词”反驳，都是直接建立在对这一公式的剖析之上的。你的考试答案应当拆解那个隐藏假定：存在是一种完善性，或者能为伟大程度增添分量。
6. Paley’s Watchmaker Analogy (Design Formula) | 佩利的手表匠类比（设计公式）
Paley’s analogical formula: If you find a watch on a heath, its complexity and purposeful arrangement of parts lead you to infer a watchmaker. Similarly, the natural world exhibits complexity and purpose (e.g., the eye) on an even greater scale, so by the same logic, we infer a divine Designer.
佩利的类比公式：如果你在荒野上发现一块手表，其复杂性和部件之间的目的性安排会让你推断出一位钟表匠。类似地，自然界在更大尺度上展现复杂性和目的（例如眼睛），因此根据同样的逻辑，我们推断出一位神圣的设计师。
Complex, purpose-ordered entities → designed by an intelligent agent. (Watch : watchmaker :: universe : God)
复杂、有目的秩序的实体 → 由智能行动者设计。（手表 : 钟表匠 :: 宇宙 : 上帝）
This formula is a posteriori and inductive. Hume’s criticisms (via Philo) – that the universe is not sufficiently like a watch, that the analogy suggests a finite, imperfect designer, or that we have no experience of universe-making – attack the strength of the inductive inference. Darwin’s alternative mechanism of natural selection offers a powerful rival explanation for apparent design.
这个公式是后天的、归纳的。休谟的批评（通过斐洛）——宇宙并不足够像一块手表，类比暗示的是一个有限的、不完美的设计师，或者我们对宇宙制造没有经验——攻击的是归纳推论的强度。达尔文的自然选择替代机制则为表面的设计提供了一个强大的竞争解释。
7. The Logical Problem of Evil | 恶的逻辑问题
The logical problem of evil is often presented as an inconsistent triad: God is omnipotent, God is wholly good, and yet evil exists. It does not invoke the quantity of evil; it simply states that the existence of any evil at all is logically incompatible with the existence of such a God.
恶的逻辑问题经常被呈现为一个不一致的三元组：上帝是全能的，上帝是全善的，而恶却存在。它不诉诸恶的数量；它只是声称，任何恶的存在在逻辑上都与这样一位上帝的存在不兼容。
Omnipotence ∧ Omnibenevolence → ¬ Evil. Evil exists. ∴ ¬ (Omnipotence ∧ Omnibenevolence).
全能 ∧ 全善 → ¬ 恶。恶存在。∴ ¬ (全能 ∧ 全善)。
Mackie’s formulation presses the theist to deny one of the premises, typically by adopting a free will defence: a wholly good God could not eliminate moral evil without eliminating free will, and free will is a greater good. Plantinga formalised this into a defence that theism is not logically inconsistent. As a ‘formula’, the triad is the starting point for all discussion of the problem of evil.
麦基的表述迫使有神论者否定其中一个前提，通常采用自由意志辩护：一个全善的上帝不能在不消除自由意志的情况下消除道德恶，而自由意志是更大的善。普兰丁格将其形式化为一个辩护，指出有神论并非逻辑上不一致。作为一个“公式”，这个三元组是所有关于恶的问题讨论的起点。
8. Aristotle’s Four Causes (Aition Schema) | 亚里士多德的四因说（四因模式）
To fully explain a thing, Aristotle provides a formula of four kinds of cause: material (that out of which it is made), formal (the essence or blueprint), efficient (the primary source of change), and final (the purpose or end, telos). This is not just for physics; it applies to ethics, where the final cause of humans is eudaimonia (flourishing).
为了充分解释一个事物，亚里士多德提供了一个四种原因（四因）的公式：质料因（构成事物的材料）、形式因（本质或蓝图）、动力因（变化的主要来源）和目的因（目的或终点，telos）。这不仅适用于物理学，也适用于伦理学，在那里，人的目的因就是 eudaimonia（幸福/繁荣）。
Complete explanation = material aition + formal aition + efficient aition + final aition.
完整的解释 = 质料因 + 形式因 + 动力因 + 目的因。
In the philosophy of mind, Aristotle’s formal account of the soul as the ‘form of a natural body that potentially has life’ is a direct application. This hylomorphic formula stands against Cartesian dualism: soul and body are not two substances but two inseparable aspects.
在心灵哲学中，亚里士多德将灵魂形式化地描述为“潜在地拥有生命的自然躯体的形式”，这正是四因的直接应用。这种形式质料论的公式与笛卡尔二元论对立：灵魂和身体不是两个实体，而是两个不可分割的方面。
9. Plato’s Theory of Forms (Formula of Participation) | 柏拉图的理型论（分有公式）
Plato’s metaphysical formula: Particular objects are what they are by ‘participating’ in the eternal, unchanging Forms (or Ideas). The Form of the Good is the highest Form, illuminating all others, like the sun in the Analogy of the Cave. Knowledge is justified true belief that grasps the Forms, not mere opinion about the changing world.
柏拉图的形而上学公式：个别事物之所以是其所是，是因为它们“分有”了永恒不变的理型（或理念）。善的理型是最高的理型，它照亮所有其他理型，就像洞穴比喻中的太阳。知识是确证了的真信念，它把握的是理型，而非关于流变世界的单纯意见。
Particular x is F iff x participates in the Form of F. (Where F is a property like ‘beauty’ or ‘justice’.)
个别物 x 是 F，当且仅当 x 分有了 F 的理型。（F 是诸如“美”或“正义”的属性。）
This formula underpins Plato’s rationalism and his argument for the immortality of the soul (learning as recollection). Key criticisms include the Third Man argument (infinite regress) and Aristotle’s question of how separate Forms can be causes of change. These are central AS topics.
这个公式支撑了柏拉图的理性主义及其对灵魂不死的论证（学习即回忆）。关键的批判包括第三人论证（无穷倒退）和亚里士多德关于分离的理型如何能成为变化之原因的问题。这些都是 AS 的核心话题。
10. Mind-Brain Identity Theory (Type Identity) | 心脑同一论（类型同一论）
The identity theory offers a clean physicalist formula: Mental states are numerically identical to brain states. Every type of mental state (e.g., pain) is identical to a type of physical state (e.g., C-fibre firing). This is not mere correlation; it is an ontological reduction.
同一论提供了一个干净利落的物理主义公式：心灵状态与大脑状态在数目上是同一的。每一类心灵状态（例如疼痛）都与某一类物理状态（例如 C-纤维放电）相同一。这不只是相关关系，而是一种本体论上的还原。
Mental state M = Physical brain state P. (Type-type identity)
心灵状态 M = 物理脑状态 P。（类型-类型同一）
This formula is contrasted with substance dualism and property dualism. The main objection is multiple realizability (Putnam): the same mental state (e.g., pain) could be realised by different neural structures in octopuses, robots, or aliens, so a strict type identity seems false. The identity theorist may retreat to token identity, altering the formula.
这个公式与实体二元论和属性二元论形成对比。主要的反对意见是多重可实现性（普特南）：同一个心灵状态（如疼痛）可以在章鱼、机器人或外星人的不同神经结构中实现，因此严格的类型同一似乎为假。同一论者可以退向个例同一，从而修改公式。
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